GDP change analysis
Sophia Wu
2021-08-01 08:36:37
*******Libraries*******
library(tswge)
library(tidyverse)
library(ggplot2)
library(tseries)
library(kableExtra)
library(knitr)
library(plotly)
library(vars)
library(vars)
library(nnfor)
library(xts)
library(GGally)
library(astsa)
library(nnfor)
library(forecast)*******Introduction*******
From Federal Reserve Economic Data, we see GDP changes every year.What is the economic cycle in the US? Are we able to predict the GDP change? Are there any significant variables causing GDP changes? Using Time series will help us analyze the important insights and predict the future trend.
*******Data Source*******
All data was downloaded from Federal Reserve Economic Data (FRED) Repository It contains 75 years quarterly data from 1947 to 2021, 6 variables listed below: 6 columns as below: Date; GDP per capita; Income receipt; gross income; Profit before tax; gdp change. Response variable is gdp change.
# import US GDP data set
df <- read.csv("~/DS 6373 Time Series/TS Project2021/data/usgdp.csv")
df %>% glimpse()## Rows: 297
## Columns: 6
## $ DATE <chr> "1/1/47", "4/1/47", "7/1/47", "10/1/47", "1/1/48", "…
## $ GDP.per.capita <int> 14203, 14101, 14008, 14161, 14316, 14495, 14515, 144…
## $ Income.receipt <dbl> 1.444, 1.560, 1.592, 1.792, 1.924, 2.024, 2.132, 2.0…
## $ gross.income <dbl> 2007.377, 2004.896, 2007.186, 2026.439, 2080.756, 21…
## $ Profit.before.tax <dbl> 30.665, 28.839, 28.431, 30.901, 32.117, 33.664, 32.8…
## $ gdp.change <dbl> -1.0, -0.8, 6.4, 6.2, 6.8, 2.3, 0.4, -5.4, -1.4, 4.2…# remove empty data, then plot dgp.change column. We may some outliers in the year 2020, which was during the COVID period.
df <- na.omit(df)
plot(df$gdp.change)
# We decided to get rid of the outlier which was affected by the COVID in the year 2020. Therefore, we deleted the data from the year 2020.
data <-head(df,-4)
plot(data$gdp.change)
# matrix of scatter plots to check all of variables' correlation. It doesn't show much correlation between gdp change and other 4 variables.However, the top two important variables are GDP.PER.CAPITA and Income.receipt with correlation of -0.2 and -0.194
head(data)## DATE GDP.per.capita Income.receipt gross.income Profit.before.tax
## 1 1/1/47 14203 1.444 2007.377 30.665
## 2 4/1/47 14101 1.560 2004.896 28.839
## 3 7/1/47 14008 1.592 2007.186 28.431
## 4 10/1/47 14161 1.792 2026.439 30.901
## 5 1/1/48 14316 1.924 2080.756 32.117
## 6 4/1/48 14495 2.024 2128.912 33.664
## gdp.change
## 1 -1.0
## 2 -0.8
## 3 6.4
## 4 6.2
## 5 6.8
## 6 2.3ggpairs(data[2:6]) #matrix of scatter plots
check data stationarity
## $autplt
## [1] 1.0000000000 0.3553734055 0.2269369119 0.0191933124 -0.0586511988
## [6] -0.1181742421 -0.0316644626 -0.0424780895 -0.0112454111 0.0823995112
## [11] 0.1032327524 0.0277494350 -0.1025853721 -0.1061111577 -0.0487059751
## [16] -0.0754354491 0.0527465538 0.0512104745 0.1079823850 0.0716927140
## [21] 0.0772534963 -0.0606065204 -0.0488685631 -0.0935929823 -0.0001411832
## [26] 0.0367856683
##
## $freq
## [1] 0.003424658 0.006849315 0.010273973 0.013698630 0.017123288 0.020547945
## [7] 0.023972603 0.027397260 0.030821918 0.034246575 0.037671233 0.041095890
## [13] 0.044520548 0.047945205 0.051369863 0.054794521 0.058219178 0.061643836
## [19] 0.065068493 0.068493151 0.071917808 0.075342466 0.078767123 0.082191781
## [25] 0.085616438 0.089041096 0.092465753 0.095890411 0.099315068 0.102739726
## [31] 0.106164384 0.109589041 0.113013699 0.116438356 0.119863014 0.123287671
## [37] 0.126712329 0.130136986 0.133561644 0.136986301 0.140410959 0.143835616
## [43] 0.147260274 0.150684932 0.154109589 0.157534247 0.160958904 0.164383562
## [49] 0.167808219 0.171232877 0.174657534 0.178082192 0.181506849 0.184931507
## [55] 0.188356164 0.191780822 0.195205479 0.198630137 0.202054795 0.205479452
## [61] 0.208904110 0.212328767 0.215753425 0.219178082 0.222602740 0.226027397
## [67] 0.229452055 0.232876712 0.236301370 0.239726027 0.243150685 0.246575342
## [73] 0.250000000 0.253424658 0.256849315 0.260273973 0.263698630 0.267123288
## [79] 0.270547945 0.273972603 0.277397260 0.280821918 0.284246575 0.287671233
## [85] 0.291095890 0.294520548 0.297945205 0.301369863 0.304794521 0.308219178
## [91] 0.311643836 0.315068493 0.318493151 0.321917808 0.325342466 0.328767123
## [97] 0.332191781 0.335616438 0.339041096 0.342465753 0.345890411 0.349315068
## [103] 0.352739726 0.356164384 0.359589041 0.363013699 0.366438356 0.369863014
## [109] 0.373287671 0.376712329 0.380136986 0.383561644 0.386986301 0.390410959
## [115] 0.393835616 0.397260274 0.400684932 0.404109589 0.407534247 0.410958904
## [121] 0.414383562 0.417808219 0.421232877 0.424657534 0.428082192 0.431506849
## [127] 0.434931507 0.438356164 0.441780822 0.445205479 0.448630137 0.452054795
## [133] 0.455479452 0.458904110 0.462328767 0.465753425 0.469178082 0.472602740
## [139] 0.476027397 0.479452055 0.482876712 0.486301370 0.489726027 0.493150685
## [145] 0.496575342 0.500000000
##
## $db
## [1] 4.53538915 4.29314162 -4.14581872 2.85385927 -2.55870129
## [6] 5.24077026 4.24669914 4.27321187 -20.03873299 -3.74767936
## [11] 7.37892047 -28.07041056 6.78863916 3.61805411 -2.49737208
## [16] -0.07458407 5.75068512 1.85826436 -0.63525231 3.07191046
## [21] 5.91810637 -4.09421398 1.00691746 1.03882233 -8.86669079
## [26] 4.97048209 3.67598448 5.34643708 1.29204389 -5.78942970
## [31] 4.68587870 6.20320433 9.84027135 4.39725913 0.87686514
## [36] 2.66170198 -3.09747403 -3.79671277 -3.58335613 -4.95225242
## [41] 0.09910420 2.48934831 0.78512431 -5.16181221 1.19749816
## [46] -14.97432955 -3.12622384 -3.67577897 0.36751321 -15.77424432
## [51] -1.47439622 3.12294814 -2.74558247 -7.85023851 1.20531546
## [56] -1.13667658 1.45668175 -2.57093883 0.49213822 0.85187253
## [61] -2.18206175 2.68042423 0.60929065 -9.65203865 -15.20043894
## [66] -15.73753791 -6.55116391 -6.12279144 -3.42543028 -4.64379653
## [71] -6.76821837 -3.74811494 -3.35972929 -5.02147456 -8.21081127
## [76] -7.94542966 -13.07819858 -18.14412007 -5.14097245 -1.61595215
## [81] -2.66235800 -2.63202541 -6.21292492 2.51237717 0.02820428
## [86] -3.25312780 -15.41217386 -4.86099153 1.99089355 -6.43719756
## [91] -0.61712396 -1.48275387 0.70281439 -6.93774146 0.30418404
## [96] -18.28103823 -16.42839072 -1.49837570 -5.39339360 -10.09861674
## [101] -4.23422577 -25.04196059 -5.35798933 3.93959346 -1.82350875
## [106] -3.89210463 -5.46611623 -5.09501754 1.74053458 -18.32549625
## [111] 1.25991910 -3.02191428 -6.15527178 -29.40994898 -12.45110252
## [116] -8.12299630 -4.27099329 -3.13555774 -2.20613474 -11.79766337
## [121] -5.01693223 -13.96455535 -1.13351520 -3.65003829 -1.13676637
## [126] -2.86179311 -9.20805139 -18.37928353 0.37370700 -0.94686937
## [131] 1.22723848 4.56840085 -6.13595670 -10.01499107 -0.61289981
## [136] -16.81300205 -14.54270675 -0.52503319 -0.37570538 -10.27933569
## [141] -4.47698374 1.22950906 -2.89308293 -7.22548197 3.68612753
## [146] -4.38893923
##
## $dbz
## [1] 2.6366117 2.6364824 2.6394242 2.6492807 2.6698485 2.7037451
## [7] 2.7515260 2.8112928 2.8788744 2.9484741 3.0135503 3.0676877
## [13] 3.1053060 3.1221721 3.1157777 3.0856887 3.0339537 2.9655872
## [19] 2.8890110 2.8161717 2.7618936 2.7420218 2.7702511 2.8542762
## [25] 2.9926748 3.1740101 3.3786486 3.5824079 3.7604678 3.8903420
## [31] 3.9535388 3.9361689 3.8289829 3.6272725 3.3309296 2.9448250
## [37] 2.4795343 1.9522626 1.3875317 0.8168001 0.2759184 -0.2002584
## [43] -0.5851551 -0.8665622 -1.0480051 -1.1445570 -1.1758246 -1.1598977
## [49] -1.1103512 -1.0361989 -0.9435914 -0.8379702 -0.7257824 -0.6153334
## [55] -0.5167745 -0.4414815 -0.4011603 -0.4069430 -0.4686065 -0.5939232
## [61] -0.7880847 -1.0531218 -1.3872690 -1.7842872 -2.2328647 -2.7163610
## [67] -3.2132860 -3.6988798 -4.1477990 -4.5372099 -4.8489935 -5.0700332
## [73] -5.1909043 -5.2047281 -5.1080295 -4.9037254 -4.6041560 -4.2314761
## [79] -3.8145152 -3.3836634 -2.9661978 -2.5835066 -2.2503044 -1.9751964
## [85] -1.7618856 -1.6105348 -1.5190072 -1.4838429 -1.5008775 -1.5654467
## [91] -1.6721554 -1.8142643 -1.9828516 -2.1660466 -2.3487802 -2.5135839
## [97] -2.6428273 -2.7222388 -2.7446967 -2.7126646 -2.6380067 -2.5392342
## [103] -2.4374875 -2.3528119 -2.3016268 -2.2954695 -2.3406328 -2.4382518
## [109] -2.5845455 -2.7711151 -2.9853797 -3.2113506 -3.4309607 -3.6260150
## [115] -3.7804963 -3.8825999 -3.9257646 -3.9083495 -3.8322973 -3.7016393
## [121] -3.5216689 -3.2990787 -3.0427200 -2.7642797 -2.4782618 -2.2010826
## [127] -1.9495320 -1.7390412 -1.5821092 -1.4870076 -1.4566986 -1.4878479
## [133] -1.5699265 -1.6846845 -1.8066924 -1.9059724 -1.9534315 -1.9283208
## [139] -1.8247980 -1.6539843 -1.4402693 -1.2141848 -1.0055242 -0.8389309
## [145] -0.7320556 -0.6952933Condition 1: Constant Mean
- Wandering and oscillating behavior
- Only one realization not knowing much
- The assumption of constant mean seems like not appear to be violated.
Condition 2: Constant Variance
It is tough to confirm whether there is constant variance since we only have one realization. It seems like there is some tendency to make big jumps at the end of the period, indicating non-constant variance over time.
Condition 3: Constant Autocorrelation
The first half of acf seems like little different from the second half of acf, but not much.
# check the 1st half of acf for condition 3
acf(data$gdp.change[1:146])
# check the 2nd half of acf for condition 3
acf(data$gdp.change[147:292])
Conclusion
- Based on the above analysis, there is some evidence showing non-stationary. But GDP change might be possibly stationary in the long term. We will analyze it using both stationary and non-stationary models.
Stationary model
# AIC selects ARMA(4,3) but BIC picks ARMA(0,2) showing ARMA(4,3) and ARMA(0,2) might be good fitting stationary models.
aic5.wge(data$gdp.change, p=0:5, q=0:5)## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of aic## p q aic
## 28 4 3 2.546052
## 23 3 4 2.556459
## 4 0 3 2.557578
## 19 3 0 2.558301
## 3 0 2 2.558863aic5.wge(data$gdp.change,type = "bic", p=0:5, q=0:5)## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of bic## p q bic
## 3 0 2 2.596638
## 7 1 0 2.597013
## 13 2 0 2.602604
## 4 0 3 2.607945
## 8 1 1 2.608552aic5.wge(data$gdp.change,type = "aic", p=0:5, q=0:5)## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of aic## p q aic
## 28 4 3 2.546052
## 23 3 4 2.556459
## 4 0 3 2.557578
## 19 3 0 2.558301
## 3 0 2 2.558863Model ID ARMA(4,3)
#aic pick ARMA(4,3), I'll test Stationary model ARMA(4,3)
ARMA43 = est.arma.wge(data$gdp.change,p=4,q=3)##
## Coefficients of Original polynomial:
## 0.8818 0.2219 -1.0225 0.2756
##
## Factor Roots Abs Recip System Freq
## 1-1.4961B+0.9788B^2 0.7643+-0.6615i 0.9893 0.1135
## 1+0.9203B -1.0866 0.9203 0.5000
## 1-0.3059B 3.2688 0.3059 0.0000
##
## ljung.wge(ARMA43$res)## Obs -0.01868356 0.06534016 0.02293769 -0.02010015 0.002663382 0.05627182 0.0008466479 -0.1152143 0.02244328 -0.007274016 0.04180263 -0.1195846 0.01192616 0.0710279 -0.02062347 0.07492956 -0.02260225 -0.006833204 -0.01724831 0.06672934 -0.03941987 0.05665657 -0.03693736 0.03576284## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 19.20712
##
## $df
## [1] 24
##
## $pval
## [1] 0.7408553ljung.wge(ARMA43$res, K = 48)## Obs -0.01868356 0.06534016 0.02293769 -0.02010015 0.002663382 0.05627182 0.0008466479 -0.1152143 0.02244328 -0.007274016 0.04180263 -0.1195846 0.01192616 0.0710279 -0.02062347 0.07492956 -0.02260225 -0.006833204 -0.01724831 0.06672934 -0.03941987 0.05665657 -0.03693736 0.03576284 0.03500223 -0.06597338 -0.005565784 0.02668376 0.08388704 -0.1041685 0.04191615 -0.03035224 0.04379716 0.06760999 -0.002961814 -0.02052359 -0.04451 -0.03807524 -0.04292932 0.03676747 -0.06629873 0.1887609 -0.02771618 0.01920448 0.05505401 -0.06396082 0.07580878 0.03925691## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 48
##
## $chi.square
## [1] 51.69399
##
## $df
## [1] 48
##
## $pval
## [1] 0.3316212f_ARMA43 = fore.arma.wge(data$gdp.change,phi = ARMA43$phi,n.ahead = 12,limits = F, lastn = T)
f_ARMA43short = fore.arma.wge(data$gdp.change,phi = ARMA43$phi,n.ahead = 2,limits = F, lastn = T)
ASE_ARMA43 = mean((data$gdp.change[(length(data$gdp.change)-11):length(data$gdp.change)] - f_ARMA43$f)^2)
ASE_ARMA43## [1] 6.879172ASE_ARMA43short = mean((data$gdp.change[(length(data$gdp.change)-1):length(data$gdp.change)] - f_ARMA43short$f)^2)
ASE_ARMA43short## [1] 37.9683#Compare Spectral Densities
sims = 5
SpecDen = parzen.wge(data$gdp.change, plot = "FALSE")
plot(SpecDen$freq,SpecDen$pzgram, type = "l", lwd = 6)
for( i in 1: sims)
{
SpecDen2 = parzen.wge(gen.arma.wge(297, phi = ARMA43$phi, plot ="FALSE"), plot = "FALSE")
lines(SpecDen2$freq,SpecDen2$pzgram, lwd = 2, col = "red")
}
#Compare ACFs
sims = 5
ACF = acf(data$gdp.change, plot = "FALSE")
plot(ACF$lag ,ACF$acf , type = "l", lwd = 6)
for( i in 1: sims)
{
ACF2 = acf(gen.arma.wge(297, phi = ARMA43$phi, plot = "FALSE"), plot = "FALSE")
lines(ACF2$lag ,ACF2$acf, lwd = 2, col = "red")
}
#Compare Generated Realizations
ARMA43gen = gen.arma.wge(297,phi = ARMA43$phi, vara = ARMA43$avar)
plotts.sample.wge(ARMA43gen)
## $autplt
## [1] 1.00000000 0.72906827 0.15425728 -0.52686538 -0.90004542 -0.86697923
## [7] -0.38076750 0.24976749 0.77694632 0.89189468 0.59430452 -0.01097494
## [13] -0.58282946 -0.88205916 -0.73188628 -0.24271599 0.36835620 0.77893529
## [19] 0.81370881 0.44420481 -0.12875717 -0.63190891 -0.81460797 -0.59721814
## [25] -0.09252516 0.44991214
##
## $freq
## [1] 0.003367003 0.006734007 0.010101010 0.013468013 0.016835017 0.020202020
## [7] 0.023569024 0.026936027 0.030303030 0.033670034 0.037037037 0.040404040
## [13] 0.043771044 0.047138047 0.050505051 0.053872054 0.057239057 0.060606061
## [19] 0.063973064 0.067340067 0.070707071 0.074074074 0.077441077 0.080808081
## [25] 0.084175084 0.087542088 0.090909091 0.094276094 0.097643098 0.101010101
## [31] 0.104377104 0.107744108 0.111111111 0.114478114 0.117845118 0.121212121
## [37] 0.124579125 0.127946128 0.131313131 0.134680135 0.138047138 0.141414141
## [43] 0.144781145 0.148148148 0.151515152 0.154882155 0.158249158 0.161616162
## [49] 0.164983165 0.168350168 0.171717172 0.175084175 0.178451178 0.181818182
## [55] 0.185185185 0.188552189 0.191919192 0.195286195 0.198653199 0.202020202
## [61] 0.205387205 0.208754209 0.212121212 0.215488215 0.218855219 0.222222222
## [67] 0.225589226 0.228956229 0.232323232 0.235690236 0.239057239 0.242424242
## [73] 0.245791246 0.249158249 0.252525253 0.255892256 0.259259259 0.262626263
## [79] 0.265993266 0.269360269 0.272727273 0.276094276 0.279461279 0.282828283
## [85] 0.286195286 0.289562290 0.292929293 0.296296296 0.299663300 0.303030303
## [91] 0.306397306 0.309764310 0.313131313 0.316498316 0.319865320 0.323232323
## [97] 0.326599327 0.329966330 0.333333333 0.336700337 0.340067340 0.343434343
## [103] 0.346801347 0.350168350 0.353535354 0.356902357 0.360269360 0.363636364
## [109] 0.367003367 0.370370370 0.373737374 0.377104377 0.380471380 0.383838384
## [115] 0.387205387 0.390572391 0.393939394 0.397306397 0.400673401 0.404040404
## [121] 0.407407407 0.410774411 0.414141414 0.417508418 0.420875421 0.424242424
## [127] 0.427609428 0.430976431 0.434343434 0.437710438 0.441077441 0.444444444
## [133] 0.447811448 0.451178451 0.454545455 0.457912458 0.461279461 0.464646465
## [139] 0.468013468 0.471380471 0.474747475 0.478114478 0.481481481 0.484848485
## [145] 0.488215488 0.491582492 0.494949495 0.498316498
##
## $db
## [1] -17.722807 -15.780379 -16.540330 -13.337904 -17.735577 -11.198522
## [7] -7.106855 -27.106045 -21.264718 -11.109951 -14.892334 -16.881045
## [13] -5.115633 -7.121839 -11.055774 -10.522675 -4.318937 -15.904894
## [19] -9.956812 -7.641727 -15.152501 -12.040951 -6.996695 -5.695905
## [25] -3.784116 -14.113067 -17.195383 -13.319145 2.865567 3.796163
## [31] 8.024147 -1.243479 6.067702 20.733269 4.602416 3.141551
## [37] 2.256744 -2.149977 -9.214650 -8.697285 -13.705560 -3.753626
## [43] -14.560679 -9.723511 -21.075288 -8.759564 -8.429120 -10.095532
## [49] -5.510713 -20.041499 -11.503451 -17.954371 -9.520929 -18.764862
## [55] -11.901209 -12.885425 -15.406435 -12.906907 -11.740363 -17.894979
## [61] -16.383956 -21.107966 -17.533291 -15.146592 -18.135168 -34.054638
## [67] -18.161405 -16.172353 -21.927401 -22.736882 -23.962232 -22.274298
## [73] -18.991745 -20.275329 -29.593486 -19.354959 -29.631677 -38.414244
## [79] -20.398834 -16.438745 -17.835019 -33.104569 -22.698102 -24.646698
## [85] -20.086335 -22.367701 -26.599003 -27.074242 -22.120462 -24.712833
## [91] -19.938057 -22.184299 -25.734010 -18.931053 -22.584082 -17.583097
## [97] -36.884985 -21.581831 -20.459641 -25.037372 -18.797039 -21.374084
## [103] -36.244720 -30.962083 -29.561485 -24.321200 -49.660321 -21.859959
## [109] -20.987812 -25.823107 -16.952282 -36.065527 -26.423265 -22.665592
## [115] -24.780264 -25.001921 -21.337345 -20.324792 -17.506920 -22.508730
## [121] -28.359951 -22.502793 -20.412486 -17.639898 -21.057668 -25.140551
## [127] -15.804550 -26.005674 -18.200561 -17.810979 -23.169273 -14.234099
## [133] -16.092081 -22.483631 -25.602457 -17.056240 -17.328225 -18.389660
## [139] -17.870707 -14.740040 -10.497871 -5.594182 -8.170840 -13.185264
## [145] -11.296566 -1.185601 -10.199828 -7.893755
##
## $dbz
## [1] -14.3195671 -14.0309637 -13.5470157 -12.8994521 -12.1592429 -11.4141080
## [7] -10.7405150 -10.1886496 -9.7796613 -9.5077107 -9.3429674 -9.2370036
## [13] -9.1347007 -8.9930481 -8.7961613 -8.5487504 -8.2406208 -7.7964506
## [19] -7.0543152 -5.8464758 -4.1634398 -2.1874529 -0.1451993 1.8060767
## [25] 3.5868744 5.1660244 6.5361911 7.7002131 8.6644820 9.4359315
## [31] 10.0207267 10.4237236 10.6482646 10.6961064 10.5673881 10.2605970
## [37] 9.7725176 9.0981698 8.2307654 7.1617686 5.8812522 4.3790039
## [43] 2.6474086 0.6883876 -1.4710652 -3.7485213 -5.9655351 -7.8537437
## [49] -9.2077944 -10.0722740 -10.6599574 -11.1285720 -11.5043219 -11.7403522
## [55] -11.8105294 -11.7628705 -11.6989358 -11.7208328 -11.8986690 -12.2647544
## [61] -12.8178907 -13.5268573 -14.3323455 -15.1537454 -15.9092191 -16.5468855
## [67] -17.0647205 -17.4975078 -17.8810620 -18.2234389 -18.5008845 -18.6788175
## [73] -18.7443306 -18.7256361 -18.6828972 -18.6818588 -18.7718380 -18.9763637
## [79] -19.2926460 -19.6950964 -20.1418592 -20.5854295 -20.9862179 -21.3228383
## [85] -21.5917091 -21.7959355 -21.9326336 -21.9891701 -21.9518788 -21.8209914
## [91] -21.6195416 -21.3887728 -21.1744255 -21.0137864 -20.9290093 -20.9263094
## [97] -20.9986337 -21.1300818 -21.3011503 -21.4938216 -21.6950216 -21.8970625
## [103] -22.0949320 -22.2820274 -22.4469526 -22.5736656 -22.6456605 -22.6524151
## [109] -22.5943253 -22.4828299 -22.3356810 -22.1704642 -21.9998006 -21.8297359
## [115] -21.6608257 -21.4904609 -21.3149563 -21.1304901 -20.9328569 -20.7167990
## [121] -20.4759889 -20.2043479 -19.8984433 -19.5597308 -19.1951130 -18.8150177
## [127] -18.4294677 -18.0434851 -17.6533011 -17.2446317 -16.7941856 -16.2752299
## [133] -15.6665201 -14.9611958 -14.1707972 -13.3220700 -12.4490260 -11.5850013
## [139] -10.7577679 -9.9879569 -9.2895920 -8.6714808 -8.1387021 -7.6938601
## [145] -7.3380342 -7.0714541 -6.8939587 -6.8052950Non-stationary model
x=data$gdp.change
#difference the data
xd1.dif=artrans.wge(x,phi.tr=1)
#xd1.dif is the differenced data
plotts.sample.wge(xd1.dif)
## $autplt
## [1] 1.00000000 -0.39535823 0.06822055 -0.10410317 -0.01189996 -0.12081320
## [7] 0.07639733 -0.03661057 -0.04446272 0.06174836 0.06867999 0.05658046
## [13] -0.10196805 -0.04498719 0.05933740 -0.12517597 0.10485179 -0.04740600
## [19] 0.06885306 -0.02859393 0.11201004 -0.11853005 0.05674837 -0.11279710
## [25] 0.03126445 0.03690878
##
## $freq
## [1] 0.003436426 0.006872852 0.010309278 0.013745704 0.017182131 0.020618557
## [7] 0.024054983 0.027491409 0.030927835 0.034364261 0.037800687 0.041237113
## [13] 0.044673540 0.048109966 0.051546392 0.054982818 0.058419244 0.061855670
## [19] 0.065292096 0.068728522 0.072164948 0.075601375 0.079037801 0.082474227
## [25] 0.085910653 0.089347079 0.092783505 0.096219931 0.099656357 0.103092784
## [31] 0.106529210 0.109965636 0.113402062 0.116838488 0.120274914 0.123711340
## [37] 0.127147766 0.130584192 0.134020619 0.137457045 0.140893471 0.144329897
## [43] 0.147766323 0.151202749 0.154639175 0.158075601 0.161512027 0.164948454
## [49] 0.168384880 0.171821306 0.175257732 0.178694158 0.182130584 0.185567010
## [55] 0.189003436 0.192439863 0.195876289 0.199312715 0.202749141 0.206185567
## [61] 0.209621993 0.213058419 0.216494845 0.219931271 0.223367698 0.226804124
## [67] 0.230240550 0.233676976 0.237113402 0.240549828 0.243986254 0.247422680
## [73] 0.250859107 0.254295533 0.257731959 0.261168385 0.264604811 0.268041237
## [79] 0.271477663 0.274914089 0.278350515 0.281786942 0.285223368 0.288659794
## [85] 0.292096220 0.295532646 0.298969072 0.302405498 0.305841924 0.309278351
## [91] 0.312714777 0.316151203 0.319587629 0.323024055 0.326460481 0.329896907
## [97] 0.333333333 0.336769759 0.340206186 0.343642612 0.347079038 0.350515464
## [103] 0.353951890 0.357388316 0.360824742 0.364261168 0.367697595 0.371134021
## [109] 0.374570447 0.378006873 0.381443299 0.384879725 0.388316151 0.391752577
## [115] 0.395189003 0.398625430 0.402061856 0.405498282 0.408934708 0.412371134
## [121] 0.415807560 0.419243986 0.422680412 0.426116838 0.429553265 0.432989691
## [127] 0.436426117 0.439862543 0.443298969 0.446735395 0.450171821 0.453608247
## [133] 0.457044674 0.460481100 0.463917526 0.467353952 0.470790378 0.474226804
## [139] 0.477663230 0.481099656 0.484536082 0.487972509 0.491408935 0.494845361
## [145] 0.498281787
##
## $db
## [1] -32.40173399 -34.70012543 -24.42062011 -17.89624349 -32.51972455
## [6] -15.97879243 -12.47433820 -13.90354174 -29.19108291 -20.70801824
## [11] -5.72543814 -23.83643217 -5.55052278 -7.25712665 -12.55498645
## [16] -9.16402519 -2.87825879 -7.42083430 -7.06191296 -7.04758619
## [21] -2.92538719 -12.33441633 -5.11539400 -5.30285598 -13.77919090
## [26] -1.29367814 -3.87497500 0.38011221 -5.23894574 -6.65341361
## [31] -0.48817880 -0.16686592 5.71872170 0.02718299 -1.56076437
## [36] -2.08452017 -5.40054421 -4.57674071 -9.18841233 -9.32890814
## [41] -1.89997175 -0.08013195 -2.32125664 -7.67704071 0.39408141
## [46] -11.74362136 -4.04140825 -3.11420587 0.68615809 -9.46310797
## [51] -0.85471750 1.46537004 -5.92688360 -6.13662032 1.07732479
## [56] -0.22746784 0.35174394 -3.34972741 0.43700597 4.31347286
## [61] -9.67156750 3.63393472 1.16154829 -6.88016445 -14.37925427
## [66] -7.78655946 -4.53193008 -5.34023898 -0.76169247 -2.20003638
## [71] -8.68463046 0.15978787 -4.73924751 -3.73101127 -6.87470943
## [76] -4.27250799 -9.76350072 -8.72823184 -9.65733994 1.20420174
## [81] -0.83584714 -0.69127899 -3.29950790 7.24300550 -1.93899229
## [86] -1.10798310 -6.05369691 3.47533710 2.06025861 2.48854590
## [91] 1.47556078 2.42936449 4.26364928 -13.49984108 3.91448201
## [96] -1.59007305 1.73379792 -7.83067134 -2.40463890 -4.03209655
## [101] 0.08700636 -4.53440303 4.19117556 7.30875697 -7.78901005
## [106] 3.08048993 -11.87567087 -7.85891172 4.10026007 -8.76326099
## [111] 6.08321359 3.14081363 -10.05025461 -7.89639972 -15.04423804
## [116] -7.03826687 0.42945128 5.49991565 0.89862504 -1.58560074
## [121] -12.05168340 -9.73760555 6.51100272 -5.46413059 -4.38161718
## [126] 4.38950167 -9.06502759 0.19543119 6.52611569 -0.13773589
## [131] 8.06583688 7.28671294 -4.99191767 -5.44553246 2.53484450
## [136] -7.16077163 -0.48427091 0.27472142 3.88740717 3.00892824
## [141] 3.94413400 4.86095892 -0.72085882 4.41973229 7.42288794
##
## $dbz
## [1] -15.2055454 -14.8969782 -14.4281010 -13.8491446 -13.2071275 -12.5389021
## [7] -11.8698379 -11.2155270 -10.5844676 -9.9806886 -9.4058922 -8.8609290
## [13] -8.3465148 -7.8631759 -7.4105306 -6.9861453 -6.5843345 -6.1953827
## [19] -5.8057560 -5.3998095 -4.9631089 -4.4866635 -3.9704944 -3.4248418
## [25] -2.8684021 -2.3245421 -1.8172136 -1.3678899 -0.9939260 -0.7080566
## [31] -0.5185213 -0.4293736 -0.4406875 -0.5485156 -0.7445524 -1.0155650
## [37] -1.3427902 -1.7017107 -2.0628529 -2.3943013 -2.6661419 -2.8558858
## [43] -2.9526868 -2.9581885 -2.8836177 -2.7448872 -2.5581319 -2.3371289
## [49] -2.0926995 -1.8334135 -1.5667972 -1.3004542 -1.0427774 -0.8031636
## [55] -0.5917884 -0.4190841 -0.2950662 -0.2286233 -0.2268262 -0.2942666
## [61] -0.4324087 -0.6389464 -0.9072118 -1.2257689 -1.5784431 -1.9451011
## [67] -2.3033832 -2.6311854 -2.9090959 -3.1216689 -3.2569238 -3.3047331
## [73] -3.2558919 -3.1035137 -2.8467034 -2.4943036 -2.0658491 -1.5886400
## [79] -1.0924647 -0.6045704 -0.1465633 0.2665587 0.6258778 0.9272892
## [85] 1.1700350 1.3553583 1.4854549 1.5628377 1.5901823 1.5706740
## [91] 1.5088032 1.4114638 1.2890832 1.1563485 1.0319267 0.9365553
## [97] 0.8892874 0.9026640 0.9786978 1.1077142 1.2707159 1.4440090
## [103] 1.6039748 1.7304959 1.8087056 1.8294935 1.7893759 1.6901831
## [109] 1.5387640 1.3466896 1.1297618 0.9070410 0.6991540 0.5259131
## [115] 0.4037261 0.3436511 0.3508449 0.4254813 0.5643960 0.7623984
## [121] 1.0125848 1.3057690 1.6297404 1.9691267 2.3062166 2.6225427
## [127] 2.9007251 3.1261356 3.2881964 3.3813746 3.4060372 3.3692768
## [133] 3.2856043 3.1770432 3.0717522 3.0001911 2.9886472 3.0518533
## [139] 3.1881812 3.3802379 3.6004484 3.8183733 4.0065589 4.1437181
## [145] 4.2157614#run aic5.wge with difference data, it selects ARMA(5,1)
aic5.wge(xd1.dif,type = "bic")## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of bic## p q bic
## 14 4 1 2.695423
## 17 5 1 2.696500
## 11 3 1 2.711637
## 2 0 1 2.741955
## 16 5 0 2.777345aic5.wge(xd1.dif,type = "aic")## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of aic## p q aic
## 17 5 1 2.608138
## 14 4 1 2.619684
## 11 3 1 2.648521
## 18 5 2 2.679915
## 16 5 0 2.701606# AIC and BIC picks p=5, q=1
fitarima=arima(data$gdp.change,order=c(5,1,1),xreg=cbind(data$GDP.per.capita,data$Income.receipt,data$gross.income))
fitarima##
## Call:
## arima(x = data$gdp.change, order = c(5, 1, 1), xreg = cbind(data$GDP.per.capita,
## data$Income.receipt, data$gross.income))
##
## Coefficients:
## ar1 ar2 ar3 ar4 ar5 ma1
## 0.1948 0.0858 -0.1064 -0.1080 -0.1133 -0.6565
## s.e. 0.1620 0.0852 0.0626 0.0683 0.0733 0.2164
## cbind(data$GDP.per.capita, data$Income.receipt, data$gross.income)1
## -0.0056
## s.e. 0.0022
## cbind(data$GDP.per.capita, data$Income.receipt, data$gross.income)2
## -0.0042
## s.e. 0.0095
## cbind(data$GDP.per.capita, data$Income.receipt, data$gross.income)3
## 0.0134
## s.e. 0.0048
##
## sigma^2 estimated as 12.14: log likelihood = -776.48, aic = 1572.96AIC(fitarima) #1718.626## [1] 1572.964Model ID ARIMA(5,1,1)
# we get ARIMA(5,1,1)
ARIMA51= est.arma.wge(xd1.dif,p = 5, q=1)##
## Coefficients of Original polynomial:
## 0.3026 0.1466 -0.0844 -0.0606 -0.0857
##
## Factor Roots Abs Recip System Freq
## 1-1.2394B+0.5441B^2 1.1388+-0.7354i 0.7376 0.0913
## 1+0.5919B -1.6895 0.5919 0.5000
## 1+0.3448B+0.2662B^2 -0.6478+-1.8269i 0.5159 0.3042
##
## ljung.wge(ARIMA51$res)## Obs 0.002634967 0.009064809 0.00849656 0.004049856 -0.003994932 0.06028995 -0.04897523 -0.08876154 0.03839984 0.06359548 0.03248956 -0.1345816 -0.06745557 0.01635845 -0.08488485 0.05570062 -0.005203244 0.03356252 0.02156046 0.07203102 -0.07917512 -0.001847757 -0.08297303 0.02357947## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 22.92076
##
## $df
## [1] 24
##
## $pval
## [1] 0.52448ljung.wge(ARIMA51$res, K = 48)## Obs 0.002634967 0.009064809 0.00849656 0.004049856 -0.003994932 0.06028995 -0.04897523 -0.08876154 0.03839984 0.06359548 0.03248956 -0.1345816 -0.06745557 0.01635845 -0.08488485 0.05570062 -0.005203244 0.03356252 0.02156046 0.07203102 -0.07917512 -0.001847757 -0.08297303 0.02357947 0.03117027 -0.04955074 -0.004059025 0.06165664 0.06731381 -0.11971 -0.01954801 -0.05669516 0.008625194 0.06457581 -0.009322939 -0.004319998 -0.02518473 -0.02670933 -0.0501059 -0.01874689 -0.09886988 0.1521388 -0.04148955 0.02224686 0.07596727 -0.0261956 0.07487633 0.02626075## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 48
##
## $chi.square
## [1] 51.92725
##
## $df
## [1] 48
##
## $pval
## [1] 0.3234607f_arima511=fore.arma.wge(data$gdp.change,phi=ARIMA51$phi,n.ahead = 12,limits = F, lastn = T)
f_arima511short=fore.arma.wge(data$gdp.change,phi=ARIMA51$phi,n.ahead = 2,limits = F, lastn = T)
ASElong = mean((data$gdp.change[(length(data$gdp.change)-11):length(data$gdp.change)] - f_arima511$f)^2)
ASElong## [1] 6.515806ASEshort = mean((data$gdp.change[(length(data$gdp.change)-1):length(data$gdp.change)] - f_arima511short$f)^2)
ASEshort## [1] 35.09099ASE = mean((data$gdp.change[(length(data$gdp.change)-11):length(data$gdp.change)] - f_arima511$f)^2)
ASE## [1] 6.515806#Compare Spectral Densities
sims = 5
SpecDen = parzen.wge(data$gdp.change, plot = "FALSE")
plot(SpecDen$freq,SpecDen$pzgram, type = "l", lwd = 6)
for( i in 1: sims)
{
SpecDen3= parzen.wge(gen.arma.wge(295, phi = ARIMA51$phi, plot ="FALSE"), plot = "FALSE")
lines(SpecDen3$freq,SpecDen3$pzgram, lwd = 2, col = "red")
}
#Compare ARIMA51 ACFs
sims = 5
ACF = acf(data$gdp.change, plot = "FALSE")
plot(ACF$lag ,ACF$acf , type = "l", lwd = 6)
for( i in 1: sims)
{
ACF2 = acf(gen.aruma.wge(319, phi = ARIMA51$phi, plot ="FALSE"), plot = "FALSE")
lines(ACF2$lag ,ACF2$acf, lwd = 2, col = "red")
}
#Compare Generated Realizations
ARIMA51gen = gen.aruma.wge(297,phi = ARIMA51$phi, vara = ARIMA51$avar)
plotts.sample.wge(ARIMA51gen)
## $autplt
## [1] 1.00000000 0.39420952 0.26526097 0.02640541 0.03210965 -0.10050527
## [7] -0.10120300 -0.02522904 -0.07175427 -0.09186705 -0.15110016 -0.12755790
## [13] -0.20458737 -0.12920343 -0.07684574 0.08215468 0.03093450 0.05443022
## [19] 0.02647287 0.04961930 0.07136296 0.10197579 0.09564815 -0.01808792
## [25] -0.01178745 -0.06741017
##
## $freq
## [1] 0.003367003 0.006734007 0.010101010 0.013468013 0.016835017 0.020202020
## [7] 0.023569024 0.026936027 0.030303030 0.033670034 0.037037037 0.040404040
## [13] 0.043771044 0.047138047 0.050505051 0.053872054 0.057239057 0.060606061
## [19] 0.063973064 0.067340067 0.070707071 0.074074074 0.077441077 0.080808081
## [25] 0.084175084 0.087542088 0.090909091 0.094276094 0.097643098 0.101010101
## [31] 0.104377104 0.107744108 0.111111111 0.114478114 0.117845118 0.121212121
## [37] 0.124579125 0.127946128 0.131313131 0.134680135 0.138047138 0.141414141
## [43] 0.144781145 0.148148148 0.151515152 0.154882155 0.158249158 0.161616162
## [49] 0.164983165 0.168350168 0.171717172 0.175084175 0.178451178 0.181818182
## [55] 0.185185185 0.188552189 0.191919192 0.195286195 0.198653199 0.202020202
## [61] 0.205387205 0.208754209 0.212121212 0.215488215 0.218855219 0.222222222
## [67] 0.225589226 0.228956229 0.232323232 0.235690236 0.239057239 0.242424242
## [73] 0.245791246 0.249158249 0.252525253 0.255892256 0.259259259 0.262626263
## [79] 0.265993266 0.269360269 0.272727273 0.276094276 0.279461279 0.282828283
## [85] 0.286195286 0.289562290 0.292929293 0.296296296 0.299663300 0.303030303
## [91] 0.306397306 0.309764310 0.313131313 0.316498316 0.319865320 0.323232323
## [97] 0.326599327 0.329966330 0.333333333 0.336700337 0.340067340 0.343434343
## [103] 0.346801347 0.350168350 0.353535354 0.356902357 0.360269360 0.363636364
## [109] 0.367003367 0.370370370 0.373737374 0.377104377 0.380471380 0.383838384
## [115] 0.387205387 0.390572391 0.393939394 0.397306397 0.400673401 0.404040404
## [121] 0.407407407 0.410774411 0.414141414 0.417508418 0.420875421 0.424242424
## [127] 0.427609428 0.430976431 0.434343434 0.437710438 0.441077441 0.444444444
## [133] 0.447811448 0.451178451 0.454545455 0.457912458 0.461279461 0.464646465
## [139] 0.468013468 0.471380471 0.474747475 0.478114478 0.481481481 0.484848485
## [145] 0.488215488 0.491582492 0.494949495 0.498316498
##
## $db
## [1] -8.32789440 0.33462075 2.22992638 -0.43816326 4.10569628
## [6] 3.67896664 -2.02712973 3.12704287 5.36775061 -2.31005486
## [11] -3.82658277 5.45720532 8.90235101 -3.59371716 10.68145478
## [16] 4.56479411 2.95978310 -0.80321509 6.73418716 3.65354190
## [21] -1.32062570 -1.89912063 5.59497349 -2.03569655 -8.98322776
## [26] 0.94393242 -1.02211724 3.84554373 1.50408952 5.30141665
## [31] -13.69280278 2.94862434 0.86998824 -0.02872617 6.98478374
## [36] 4.36234726 -0.08005701 -8.00011805 -9.16376709 2.82479159
## [41] 4.64063966 5.74741288 4.27404715 -8.36936946 -7.87247854
## [46] -3.09629275 -10.87123040 -4.79680793 -7.57990033 -4.60045897
## [51] -4.47785535 -8.22651624 -17.56914236 4.35237830 -0.31551557
## [56] -6.48781054 -0.50969966 -2.54042289 0.86069823 1.14260393
## [61] -11.74527972 0.92766916 -8.51917702 -3.21031842 -9.46488118
## [66] -10.06089020 -1.83716932 -1.08199097 -13.50380302 0.09506812
## [71] -3.07110316 -0.76776432 -7.87296815 1.59755783 -10.11947248
## [76] -1.72182415 -10.02703638 1.68374722 -2.23957451 -0.87971353
## [81] -13.11166067 5.40538896 -12.75059997 -8.88561296 -15.21517969
## [86] 2.52362184 -2.73323946 -7.39245813 -4.58262355 -11.55110954
## [91] -6.34526564 -13.82476275 -6.86216590 -2.17674049 -1.28368750
## [96] -4.35577621 -8.11389188 -11.53252706 -4.77938733 -5.95942263
## [101] -1.19214184 -7.25401990 -1.90651837 -22.57952057 -12.21754378
## [106] -17.85430582 -2.64588923 -3.87826677 -12.99872118 -7.22028860
## [111] -4.62925774 -6.89560989 -0.91674656 -2.24534967 -10.35972262
## [116] -12.33113033 -16.33225775 -5.41516075 1.39321235 -2.54798050
## [121] -18.40180472 -6.66681388 -2.59132767 -6.47198476 1.75159376
## [126] -8.19558467 -0.18600938 -13.84129982 -13.97720035 -3.13983960
## [131] -8.08662879 0.80789846 0.63040592 -9.27598601 -3.74044750
## [136] 0.44529556 5.47844823 1.21170927 -5.63280609 -12.34310698
## [141] -0.94050443 -0.40943426 -0.20102961 -0.33029281 -0.85293176
## [146] -5.91130845 -10.57635130 -3.51355011
##
## $dbz
## [1] 0.9866247 1.1493266 1.4074737 1.7436188 2.1367815 2.5646232
## [7] 3.0051688 3.4379712 3.8447834 4.2098701 4.5200867 4.7648336
## [13] 4.9359676 5.0277392 5.0368085 4.9623923 4.8065824 4.5748617
## [19] 4.2768001 3.9268169 3.5447261 3.1555375 2.7878060 2.4700064
## [25] 2.2252808 2.0662843 1.9925999 1.9921613 2.0458158 2.1326106
## [31] 2.2337024 2.3341478 2.4229390 2.4920463 2.5351387 2.5464322
## [37] 2.5198997 2.4489316 2.3264319 2.1452934 1.8991953 1.5837138
## [43] 1.1978092 0.7457990 0.2398742 -0.2971123 -0.8297003 -1.3111708
## [49] -1.6913871 -1.9317304 -2.0207699 -1.9796422 -1.8526558 -1.6906633
## [55] -1.5377631 -1.4254276 -1.3721753 -1.3855864 -1.4646151 -1.6014332
## [61] -1.7827822 -1.9911345 -2.2060618 -2.4061421 -2.5714739 -2.6864292
## [67] -2.7418702 -2.7360174 -2.6736499 -2.5640666 -2.4187021 -2.2492160
## [73] -2.0664407 -1.8801307 -1.6992081 -1.5321600 -1.3873299 -1.2729720
## [79] -1.1970560 -1.1668763 -1.1885443 -1.2664290 -1.4025797 -1.5961522
## [85] -1.8428656 -2.1345657 -2.4590603 -2.8004923 -3.1405674 -3.4608184
## [91] -3.7456768 -3.9855358 -4.1786323 -4.3308883 -4.4537810 -4.5611779
## [97] -4.6662818 -4.7794121 -4.9067637 -5.0499276 -5.2058860 -5.3673178
## [103] -5.5232266 -5.6600488 -5.7634051 -5.8204558 -5.8224286 -5.7665227
## [109] -5.6564108 -5.5010795 -5.3124922 -5.1029927 -4.8832849 -4.6614063
## [115] -4.4426821 -4.2303832 -4.0266963 -3.8336118 -3.6534086 -3.4885509
## [121] -3.3409939 -3.2110674 -3.0962389 -2.9901550 -2.8824274 -2.7596298
## [127] -2.6077637 -2.4158982 -2.1798930 -1.9046499 -1.6038028 -1.2970306
## [133] -1.0062788 -0.7523259 -0.5524852 -0.4194828 -0.3611469 -0.3804836
## [139] -0.4758399 -0.6410117 -0.8652989 -1.1336244 -1.4269169 -1.7229890
## [145] -1.9980670 -2.2289362 -2.3953992 -2.4825701Model ID ARUMA(5,1,0) S=27
# overfit for seasonality, we get system frequency 0.02 associated with period 27.
plotts.sample.wge(x)
## $autplt
## [1] 1.0000000000 0.3553734055 0.2269369119 0.0191933124 -0.0586511988
## [6] -0.1181742421 -0.0316644626 -0.0424780895 -0.0112454111 0.0823995112
## [11] 0.1032327524 0.0277494350 -0.1025853721 -0.1061111577 -0.0487059751
## [16] -0.0754354491 0.0527465538 0.0512104745 0.1079823850 0.0716927140
## [21] 0.0772534963 -0.0606065204 -0.0488685631 -0.0935929823 -0.0001411832
## [26] 0.0367856683
##
## $freq
## [1] 0.003424658 0.006849315 0.010273973 0.013698630 0.017123288 0.020547945
## [7] 0.023972603 0.027397260 0.030821918 0.034246575 0.037671233 0.041095890
## [13] 0.044520548 0.047945205 0.051369863 0.054794521 0.058219178 0.061643836
## [19] 0.065068493 0.068493151 0.071917808 0.075342466 0.078767123 0.082191781
## [25] 0.085616438 0.089041096 0.092465753 0.095890411 0.099315068 0.102739726
## [31] 0.106164384 0.109589041 0.113013699 0.116438356 0.119863014 0.123287671
## [37] 0.126712329 0.130136986 0.133561644 0.136986301 0.140410959 0.143835616
## [43] 0.147260274 0.150684932 0.154109589 0.157534247 0.160958904 0.164383562
## [49] 0.167808219 0.171232877 0.174657534 0.178082192 0.181506849 0.184931507
## [55] 0.188356164 0.191780822 0.195205479 0.198630137 0.202054795 0.205479452
## [61] 0.208904110 0.212328767 0.215753425 0.219178082 0.222602740 0.226027397
## [67] 0.229452055 0.232876712 0.236301370 0.239726027 0.243150685 0.246575342
## [73] 0.250000000 0.253424658 0.256849315 0.260273973 0.263698630 0.267123288
## [79] 0.270547945 0.273972603 0.277397260 0.280821918 0.284246575 0.287671233
## [85] 0.291095890 0.294520548 0.297945205 0.301369863 0.304794521 0.308219178
## [91] 0.311643836 0.315068493 0.318493151 0.321917808 0.325342466 0.328767123
## [97] 0.332191781 0.335616438 0.339041096 0.342465753 0.345890411 0.349315068
## [103] 0.352739726 0.356164384 0.359589041 0.363013699 0.366438356 0.369863014
## [109] 0.373287671 0.376712329 0.380136986 0.383561644 0.386986301 0.390410959
## [115] 0.393835616 0.397260274 0.400684932 0.404109589 0.407534247 0.410958904
## [121] 0.414383562 0.417808219 0.421232877 0.424657534 0.428082192 0.431506849
## [127] 0.434931507 0.438356164 0.441780822 0.445205479 0.448630137 0.452054795
## [133] 0.455479452 0.458904110 0.462328767 0.465753425 0.469178082 0.472602740
## [139] 0.476027397 0.479452055 0.482876712 0.486301370 0.489726027 0.493150685
## [145] 0.496575342 0.500000000
##
## $db
## [1] 4.53538915 4.29314162 -4.14581872 2.85385927 -2.55870129
## [6] 5.24077026 4.24669914 4.27321187 -20.03873299 -3.74767936
## [11] 7.37892047 -28.07041056 6.78863916 3.61805411 -2.49737208
## [16] -0.07458407 5.75068512 1.85826436 -0.63525231 3.07191046
## [21] 5.91810637 -4.09421398 1.00691746 1.03882233 -8.86669079
## [26] 4.97048209 3.67598448 5.34643708 1.29204389 -5.78942970
## [31] 4.68587870 6.20320433 9.84027135 4.39725913 0.87686514
## [36] 2.66170198 -3.09747403 -3.79671277 -3.58335613 -4.95225242
## [41] 0.09910420 2.48934831 0.78512431 -5.16181221 1.19749816
## [46] -14.97432955 -3.12622384 -3.67577897 0.36751321 -15.77424432
## [51] -1.47439622 3.12294814 -2.74558247 -7.85023851 1.20531546
## [56] -1.13667658 1.45668175 -2.57093883 0.49213822 0.85187253
## [61] -2.18206175 2.68042423 0.60929065 -9.65203865 -15.20043894
## [66] -15.73753791 -6.55116391 -6.12279144 -3.42543028 -4.64379653
## [71] -6.76821837 -3.74811494 -3.35972929 -5.02147456 -8.21081127
## [76] -7.94542966 -13.07819858 -18.14412007 -5.14097245 -1.61595215
## [81] -2.66235800 -2.63202541 -6.21292492 2.51237717 0.02820428
## [86] -3.25312780 -15.41217386 -4.86099153 1.99089355 -6.43719756
## [91] -0.61712396 -1.48275387 0.70281439 -6.93774146 0.30418404
## [96] -18.28103823 -16.42839072 -1.49837570 -5.39339360 -10.09861674
## [101] -4.23422577 -25.04196059 -5.35798933 3.93959346 -1.82350875
## [106] -3.89210463 -5.46611623 -5.09501754 1.74053458 -18.32549625
## [111] 1.25991910 -3.02191428 -6.15527178 -29.40994898 -12.45110252
## [116] -8.12299630 -4.27099329 -3.13555774 -2.20613474 -11.79766337
## [121] -5.01693223 -13.96455535 -1.13351520 -3.65003829 -1.13676637
## [126] -2.86179311 -9.20805139 -18.37928353 0.37370700 -0.94686937
## [131] 1.22723848 4.56840085 -6.13595670 -10.01499107 -0.61289981
## [136] -16.81300205 -14.54270675 -0.52503319 -0.37570538 -10.27933569
## [141] -4.47698374 1.22950906 -2.89308293 -7.22548197 3.68612753
## [146] -4.38893923
##
## $dbz
## [1] 2.6366117 2.6364824 2.6394242 2.6492807 2.6698485 2.7037451
## [7] 2.7515260 2.8112928 2.8788744 2.9484741 3.0135503 3.0676877
## [13] 3.1053060 3.1221721 3.1157777 3.0856887 3.0339537 2.9655872
## [19] 2.8890110 2.8161717 2.7618936 2.7420218 2.7702511 2.8542762
## [25] 2.9926748 3.1740101 3.3786486 3.5824079 3.7604678 3.8903420
## [31] 3.9535388 3.9361689 3.8289829 3.6272725 3.3309296 2.9448250
## [37] 2.4795343 1.9522626 1.3875317 0.8168001 0.2759184 -0.2002584
## [43] -0.5851551 -0.8665622 -1.0480051 -1.1445570 -1.1758246 -1.1598977
## [49] -1.1103512 -1.0361989 -0.9435914 -0.8379702 -0.7257824 -0.6153334
## [55] -0.5167745 -0.4414815 -0.4011603 -0.4069430 -0.4686065 -0.5939232
## [61] -0.7880847 -1.0531218 -1.3872690 -1.7842872 -2.2328647 -2.7163610
## [67] -3.2132860 -3.6988798 -4.1477990 -4.5372099 -4.8489935 -5.0700332
## [73] -5.1909043 -5.2047281 -5.1080295 -4.9037254 -4.6041560 -4.2314761
## [79] -3.8145152 -3.3836634 -2.9661978 -2.5835066 -2.2503044 -1.9751964
## [85] -1.7618856 -1.6105348 -1.5190072 -1.4838429 -1.5008775 -1.5654467
## [91] -1.6721554 -1.8142643 -1.9828516 -2.1660466 -2.3487802 -2.5135839
## [97] -2.6428273 -2.7222388 -2.7446967 -2.7126646 -2.6380067 -2.5392342
## [103] -2.4374875 -2.3528119 -2.3016268 -2.2954695 -2.3406328 -2.4382518
## [109] -2.5845455 -2.7711151 -2.9853797 -3.2113506 -3.4309607 -3.6260150
## [115] -3.7804963 -3.8825999 -3.9257646 -3.9083495 -3.8322973 -3.7016393
## [121] -3.5216689 -3.2990787 -3.0427200 -2.7642797 -2.4782618 -2.2010826
## [127] -1.9495320 -1.7390412 -1.5821092 -1.4870076 -1.4566986 -1.4878479
## [133] -1.5699265 -1.6846845 -1.8066924 -1.9059724 -1.9534315 -1.9283208
## [139] -1.8247980 -1.6539843 -1.4402693 -1.2141848 -1.0055242 -0.8389309
## [145] -0.7320556 -0.6952933est.ar.wge(data$gdp.change,p = 12) ##
## Coefficients of Original polynomial:
## 0.3141 0.1840 -0.0661 -0.0679 -0.0820 0.0904 -0.0460 -0.0585 0.0865 0.0954 -0.0026 -0.1755
##
## Factor Roots Abs Recip System Freq
## 1-1.4059B+0.8259B^2 0.8511+-0.6974i 0.9088 0.1093
## 1-0.5698B+0.7862B^2 0.3624+-1.0680i 0.8867 0.1979
## 1+0.5302B+0.7457B^2 -0.3555+-1.1021i 0.8636 0.2997
## 1-1.6762B+0.7418B^2 1.1298+-0.2675i 0.8613 0.0370
## 1+1.6592B+0.7279B^2 -1.1397+-0.2736i 0.8532 0.4625
## 1+1.1485B+0.6711B^2 -0.8556+-0.8706i 0.8192 0.3736
##
##
##
## Coefficients of Original polynomial:
## 0.3141 0.1840 -0.0661 -0.0679 -0.0820 0.0904 -0.0460 -0.0585 0.0865 0.0954 -0.0026 -0.1755
##
## Factor Roots Abs Recip System Freq
## 1-1.4059B+0.8259B^2 0.8511+-0.6974i 0.9088 0.1093
## 1-0.5698B+0.7862B^2 0.3624+-1.0680i 0.8867 0.1979
## 1+0.5302B+0.7457B^2 -0.3555+-1.1021i 0.8636 0.2997
## 1-1.6762B+0.7418B^2 1.1298+-0.2675i 0.8613 0.0370
## 1+1.6592B+0.7279B^2 -1.1397+-0.2736i 0.8532 0.4625
## 1+1.1485B+0.6711B^2 -0.8556+-0.8706i 0.8192 0.3736
##
## ## $phi
## [1] 0.314069511 0.184020866 -0.066080984 -0.067940844 -0.081972592
## [6] 0.090375734 -0.045990923 -0.058470179 0.086480690 0.095363038
## [11] -0.002610369 -0.175483206
##
## $res
## [1] -2.15359860 -2.07598972 4.23624661 1.79039761 1.02353242 -2.59234854
## [7] -2.52389829 -6.29125812 -1.00432837 4.14738663 -6.40877986 13.91327758
## [13] 4.72957860 6.28154026 -0.50476007 -0.51268863 6.96319858 5.25307814
## [19] -3.90258753 0.64451344 -2.01239257 -0.41655565 8.34875561 1.43868167
## [25] -1.75209743 -3.79529432 -6.26008643 0.02062483 0.20427104 3.34657360
## [31] 4.26806960 5.68030889 -1.86271033 -0.90538414 0.70541720 -2.11294224
## [37] 3.57603914 -3.08341258 3.35660956 -2.39808811 -6.48191956 1.85628667
## [43] -6.80108196 -9.62375566 5.35581744 9.08248080 3.99075137 -1.11256149
## [49] 2.93243839 -4.14945867 -1.07448630 7.62897837 -5.40862331 1.19273737
## [55] -8.58675150 -0.46971039 4.85709866 1.90229789 3.50952359 2.77606147
## [61] -0.03529566 0.81795689 -2.14277705 3.79442697 1.76400122 4.90861290
## [67] -4.37739213 3.78599050 0.32209457 2.61442603 -1.92002065 7.57255579
## [73] 1.46820425 4.24597073 3.79182612 4.04661133 -3.78797120 0.37607765
## [79] 1.30093701 1.99159199 -3.03351974 1.11647509 0.04304231 5.38807081
## [85] 1.03346292 -1.87547307 -0.28225566 5.32662989 -2.51406124 -0.44265376
## [91] -4.31979083 -1.63173864 -0.95909353 0.41591247 -7.83380618 10.70915943
## [97] -1.77674586 -1.80422894 -2.33563168 5.65442250 6.86116176 -2.92633946
## [103] 1.85807568 6.69958197 -1.17026024 -7.29034331 1.14502979 -2.83708097
## [109] -0.19021689 -6.58251807 -3.54402543 -4.42358629 1.84061367 4.06966639
## [115] 0.70497794 5.58356127 -2.26031903 -1.66486617 1.21539433 1.55826179
## [121] 5.47311059 1.65100721 -6.09687349 -2.77993673 14.31496060 -2.52244964
## [127] -0.24837444 -1.76643177 -1.00726180 1.97549566 -3.87925331 -0.73970535
## [133] -8.66399765 -0.05389442 5.81581254 2.29217607 -7.27199878 2.56005856
## [139] -5.09285526 -7.47239531 1.66579532 -2.75762252 -0.19950060 2.13600978
## [145] 2.47915634 2.04798482 2.75716864 3.75892737 2.74063388 0.46799637
## [151] -1.15726831 0.62167030 0.83168250 1.65341146 -2.44640284 0.06869586
## [157] -0.84060130 1.33661824 -0.14568867 0.51467019 2.25744722 -0.02490905
## [163] 3.34010714 -2.58769834 2.03179010 -0.30116241 2.13452414 0.89086160
## [169] -1.15701518 0.27911131 -2.50432000 1.96012748 -1.72962665 -2.80230032
## [175] -4.95938944 -2.87865185 2.95724366 -1.47485726 -1.96138918 2.23532204
## [181] 1.00348249 -0.10564480 -0.20295620 -2.01830914 0.88073042 -0.67065702
## [187] 1.71860553 -0.48975332 1.33931724 -1.80243023 0.97525107 -1.64429013
## [193] -1.53101808 1.93773789 0.03778953 -0.52058390 3.01731963 -1.15837150
## [199] 0.66252625 -0.93857576 4.22994873 1.50359993 -0.82388938 0.56121747
## [205] 0.45111059 1.88735350 2.18253299 -0.86892833 0.20076723 2.30089881
## [211] 3.27097010 -3.34685307 4.67595792 -2.92763966 -0.24171375 -3.56346497
## [217] 0.14983451 -2.97503984 -0.77063866 1.39208744 -1.26061388 -1.35925765
## [223] -2.20344290 0.06984580 2.08877217 3.16542479 0.23784868 -2.41741783
## [229] 0.28722431 0.27619794 0.69650660 0.99086331 -1.66118634 0.87432198
## [235] -1.04080098 1.62164555 -2.87474245 -1.65305990 2.14010799 -2.28347409
## [241] -0.58348991 -0.73167478 -0.02565988 -4.77625103 -0.18614761 -3.98829425
## [247] -9.87353305 -2.95395180 -0.41498137 -0.06196149 0.84761052 -3.64232649
## [253] 0.93804298 -0.08479094 -1.63615567 -3.67242162 2.27477053 -2.26300405
## [259] 0.63747793 -1.60489213 -2.86379206 -2.22721261 -1.77277310 1.64809692
## [265] -2.33438759 0.56855061 0.44895754 -5.03566138 3.34239206 0.89866501
## [271] -1.41369832 0.81225515 -0.79063530 -1.28419009 -2.45420994 -0.06709437
## [277] -1.02462776 -0.33044649 -0.76311446 -1.55791481 -0.77294672 0.24339743
## [283] 0.91259213 0.67374449 -0.85827674 -1.20004182 -1.52287584 0.47748239
## [289] -1.65510830 -0.23877178 -0.36557741 -8.19436336
##
## $avar
## [1] 11.73459
##
## $aic
## [1] 2.551582
##
## $aicc
## [1] 3.563624
##
## $bic
## [1] 2.715273factor.wge(data$gdp.change)##
## Coefficients of Original polynomial:
## -1.0000 -0.8000 6.4000 6.2000 6.8000 2.3000 0.4000 -5.4000 -1.4000 4.2000 -3.3000 16.7000 12.8000 16.4000 7.9000 5.5000 7.1000 8.5000 0.9000 4.3000 0.9000 2.9000 13.8000 7.6000 3.1000 -2.2000 -5.9000 -1.9000 0.4000 4.6000 8.1000 11.9000 6.7000 5.5000 2.4000 -1.5000 3.3000 -0.4000 6.7000 2.6000 -0.9000 4.0000 -4.1000 -10.0000 2.7000 9.6000 9.7000 7.9000 9.3000 0.3000 1.1000 9.3000 -2.1000 2.0000 -5.0000 2.7000 7.0000 7.9000 8.1000 7.3000 3.7000 5.0000 1.3000 4.4000 4.6000 9.1000 2.6000 8.7000 4.4000 6.4000 1.2000 10.0000 5.1000 9.2000 9.5000 10.1000 1.4000 3.4000 3.3000 3.6000 0.2000 3.8000 3.0000 8.4000 6.9000 3.1000 1.6000 6.4000 1.2000 2.7000 -1.9000 -0.6000 0.6000 3.7000 -4.2000 11.3000 2.2000 3.3000 0.9000 7.6000 9.4000 3.8000 6.9000 10.3000 4.4000 -2.1000 3.8000 -3.4000 1.0000 -3.7000 -1.5000 -4.8000 2.9000 7.0000 5.5000 9.3000 3.0000 2.2000 2.9000 4.8000 8.0000 7.4000 0.0000 1.3000 16.4000 4.1000 5.5000 0.7000 0.4000 3.0000 1.0000 1.3000 -8.0000 -0.5000 7.7000 8.1000 -2.9000 4.9000 -4.3000 -6.1000 1.8000 -1.5000 0.2000 5.4000 9.4000 8.2000 8.6000 8.1000 7.1000 3.9000 3.3000 3.9000 3.6000 6.2000 3.0000 3.8000 1.8000 3.9000 2.2000 3.0000 4.4000 3.5000 7.0000 2.1000 5.4000 2.4000 5.4000 4.1000 3.1000 3.0000 0.8000 4.4000 1.5000 0.3000 -3.6000 -1.9000 3.2000 2.0000 1.4000 4.9000 4.4000 4.0000 4.2000 0.7000 2.3000 1.9000 5.6000 3.9000 5.5000 2.4000 4.7000 1.4000 1.2000 3.5000 2.7000 3.0000 6.8000 3.6000 4.2000 2.6000 6.8000 5.1000 3.5000 4.1000 3.8000 5.1000 6.6000 3.8000 3.1000 5.3000 7.0000 1.5000 7.5000 0.5000 2.5000 -1.1000 2.4000 -1.7000 1.1000 3.5000 2.4000 1.8000 0.6000 2.2000 3.5000 7.0000 4.7000 2.2000 3.1000 3.8000 4.1000 4.5000 1.9000 3.6000 2.6000 5.4000 0.9000 0.6000 3.5000 0.9000 2.3000 2.2000 2.5000 -2.3000 2.1000 -2.1000 -8.4000 -4.4000 -0.6000 1.5000 4.5000 1.5000 3.7000 3.0000 2.0000 -1.0000 2.9000 -0.1000 4.7000 3.2000 1.7000 0.5000 0.5000 3.6000 0.5000 3.2000 3.2000 -1.1000 5.5000 5.0000 2.3000 3.8000 2.7000 1.5000 0.6000 2.3000 1.3000 2.2000 2.5000 2.3000 1.7000 2.9000 3.9000 3.8000 2.7000 2.1000 1.3000 2.9000 1.5000 2.6000 2.4000 -5.0000
##
## Factor Roots Abs Recip System Freq
## 1-1.8896B 0.5292 1.8896 0.0000
## 1+1.9433B+3.5320B^2 -0.2751+-0.4555i 1.8794 0.3365
## 1+1.9422B+1.6237B^2 -0.5981+-0.5081i 1.2742 0.3879
## 1+0.0375B+1.2902B^2 -0.0145+-0.8803i 1.1359 0.2526
## 1+1.1327B -0.8828 1.1327 0.5000
## 1-0.9560B+1.1640B^2 0.4106+-0.8309i 1.0789 0.1769
## 1+0.1251B+1.0567B^2 -0.0592+-0.9710i 1.0280 0.2597
## 1+0.8970B+1.0497B^2 -0.4273+-0.8775i 1.0246 0.3221
## 1+0.5467B+1.0462B^2 -0.2613+-0.9421i 1.0228 0.2931
## 1+1.0203B -0.9769 1.0220 0.4909
## 1+1.0203B -0.9769 1.0220 0.4909
## 1+1.7854B+1.0425B^2 -0.8564+-0.4753i 1.0210 0.4194
## 1-1.3765B+1.0412B^2 0.6610+-0.7235i 1.0204 0.1322
## 1+0.8316B+1.0397B^2 -0.3999+-0.8955i 1.0197 0.3168
## 1+1.1049B+1.0386B^2 -0.5319+-0.8245i 1.0191 0.3412
## 1-0.0523B+1.0363B^2 0.0252+-0.9820i 1.0180 0.2459
## 1-0.1960B+1.0325B^2 0.0949+-0.9795i 1.0161 0.2346
## 1-0.7228B+1.0309B^2 0.3506+-0.9204i 1.0153 0.1921
## 1+0.4165B+1.0132B^2 -0.2055+-0.9720i 1.0066 0.2832
## 1-1.0839B+1.0119B^2 0.5356+-0.8375i 1.0059 0.1594
## 1+1.5355B+1.0112B^2 -0.7592+-0.6422i 1.0056 0.3883
## 1-0.3884B+1.0110B^2 0.1921+-0.9758i 1.0055 0.2191
## 1+1.8976B+1.0105B^2 -0.9389+-0.3286i 1.0053 0.4464
## 1-0.6185B+1.0095B^2 0.3063+-0.9470i 1.0047 0.2002
## 1+0.9673B -0.9583 1.0047 0.4564
## 1+0.9673B -0.9583 1.0047 0.4564
## 1+0.9850B -0.9786 1.0033 0.4696
## 1+0.9850B -0.9786 1.0033 0.4696
## 1+0.8669B -0.8617 1.0030 0.4161
## 1+0.8669B -0.8617 1.0030 0.4161
## 1-0.6917B 0.6877 1.0029 0.1289
## 1-0.6917B 0.6877 1.0029 0.1289
## 1+1.4401B+1.0054B^2 -0.7162+-0.6941i 1.0027 0.3775
## 1+0.9941B -0.9888 1.0026 0.4792
## 1-0.5512B 0.5483 1.0026 0.1574
## 1-0.5512B 0.5484 1.0026 0.1574
## 1+0.9939B -0.9890 1.0025 0.4792
## 1+0.9282B -0.9239 1.0023 0.4384
## 1+0.9282B -0.9239 1.0023 0.4384
## 1-0.5024B+1.0040B^2 0.2502+-0.9661i 1.0020 0.2097
## 1-0.3086B+1.0030B^2 0.1539+-0.9866i 1.0015 0.2254
## 1-0.9502B+1.0029B^2 0.4737+-0.8790i 1.0015 0.1713
## 1+0.5890B+1.0026B^2 -0.2937+-0.9545i 1.0013 0.2975
## 1-0.7074B 0.7058 1.0011 0.1251
## 1+1.1707B+1.0017B^2 -0.5843+-0.8104i 1.0009 0.3494
## 1+0.7053B+1.0001B^2 -0.3526+-0.9357i 1.0000 0.3074
## 1+1.3455B+0.9993B^2 -0.6732+-0.7399i 0.9997 0.3675
## 1+1.7848B+0.9993B^2 -0.8930+-0.4508i 0.9996 0.4256
## 1+1.5415B+0.9992B^2 -0.7714+-0.6370i 0.9996 0.3901
## 1+1.8385B+0.9989B^2 -0.9202+-0.3928i 0.9995 0.4358
## 1+0.9679B -0.9689 0.9995 0.4599
## 1+0.9678B -0.9689 0.9994 0.4599
## 1+1.0414B+0.9987B^2 -0.5214+-0.8541i 0.9993 0.3372
## 1+0.9475B+0.9985B^2 -0.4745+-0.8811i 0.9992 0.3286
## 1+0.9847B+0.9984B^2 -0.4932+-0.8709i 0.9992 0.3320
## 1+1.8081B+0.9978B^2 -0.9061+-0.4257i 0.9989 0.4301
## 1+1.2852B+0.9969B^2 -0.6446+-0.7665i 0.9985 0.3613
## 1+0.8832B+0.9966B^2 -0.4431+-0.8984i 0.9983 0.3229
## 1-0.8077B+0.9964B^2 0.4053+-0.9162i 0.9982 0.1837
## 1+1.3756B+0.9963B^2 -0.6903+-0.7261i 0.9981 0.3710
## 1-0.5647B+0.9961B^2 0.2834+-0.9610i 0.9980 0.2044
## 1+0.2185B+0.9959B^2 -0.1097+-0.9960i 0.9980 0.2675
## 1+0.2745B+0.9954B^2 -0.1379+-0.9928i 0.9977 0.2720
## 1+0.8469B -0.8509 0.9977 0.4114
## 1+0.8469B -0.8509 0.9977 0.4114
## 1-0.7082B 0.7116 0.9976 0.1244
## 1+0.8708B -0.8751 0.9975 0.4189
## 1+0.8708B -0.8751 0.9975 0.4189
## 1-1.1514B+0.9946B^2 0.5788+-0.8188i 0.9973 0.1521
## 1+0.9748B -0.9801 0.9973 0.4661
## 1+0.9748B -0.9801 0.9973 0.4661
## 1-0.3349B+0.9944B^2 0.1684+-0.9886i 0.9972 0.2231
## 1+1.4257B+0.9934B^2 -0.7176+-0.7012i 0.9967 0.3768
## 1+0.6217B+0.9923B^2 -0.3133+-0.9538i 0.9961 0.3005
## 1+0.9958B -1.0042 0.9958 0.5000
## 1+1.5743B+0.9915B^2 -0.7939+-0.6150i 0.9958 0.3951
## 1-1.5278B+0.9914B^2 0.7705+-0.6442i 0.9957 0.1108
## 1+1.6047B+0.9910B^2 -0.8096+-0.5946i 0.9955 0.3992
## 1+0.8171B -0.8249 0.9953 0.4033
## 1+0.8171B -0.8249 0.9953 0.4033
## 1+0.9825B -0.9920 0.9952 0.4746
## 1+0.9825B -0.9921 0.9952 0.4746
## 1-0.6581B 0.6646 0.9950 0.1350
## 1-0.7392B 0.7467 0.9950 0.1167
## 1+0.1777B+0.9898B^2 -0.0898+-1.0011i 0.9949 0.2642
## 1-0.5889B 0.5951 0.9947 0.1492
## 1+0.0183B+0.9895B^2 -0.0092+-1.0053i 0.9947 0.2515
## 1+0.5267B+0.9892B^2 -0.2662+-0.9696i 0.9946 0.2926
## 1-0.5886B 0.5952 0.9945 0.1492
## 1-0.6562B 0.6638 0.9942 0.1353
## 1-1.0274B+0.9884B^2 0.5197+-0.8612i 0.9942 0.1636
## 1+0.5974B -0.6047 0.9940 0.3526
## 1+0.5974B -0.6047 0.9940 0.3526
## 1-1.2816B+0.9879B^2 0.6487+-0.7691i 0.9939 0.1385
## 1+1.4765B+0.9871B^2 -0.7478+-0.6736i 0.9935 0.3833
## 1-0.0698B+0.9869B^2 0.0353+-1.0060i 0.9934 0.2444
## 1-1.4544B+0.9868B^2 0.7369+-0.6858i 0.9934 0.1193
## 1+0.9873B -1.0014 0.9930 0.4831
## 1+0.9873B -1.0015 0.9929 0.4831
## 1+0.2217B -0.2250 0.9927 0.2859
## 1+0.2217B -0.2250 0.9927 0.2859
## 1+1.0855B+0.9848B^2 -0.5511+-0.8436i 0.9924 0.3421
## 1-0.9021B+0.9848B^2 0.4580+-0.8976i 0.9924 0.1749
## 1+0.8316B -0.8445 0.9924 0.4081
## 1+0.8316B -0.8445 0.9923 0.4081
## 1-0.9877B+0.9814B^2 0.5032+-0.8751i 0.9906 0.1669
## 1-0.2484B+0.9812B^2 0.1266+-1.0016i 0.9906 0.2300
## 1+0.9895B -1.0087 0.9905 0.4932
## 1-0.6443B+0.9809B^2 0.3284+-0.9548i 0.9904 0.1973
## 1+0.9894B -1.0088 0.9903 0.4932
## 1-0.0625B 0.0637 0.9902 0.2399
## 1-0.0625B 0.0637 0.9902 0.2399
## 1-0.7381B 0.7529 0.9901 0.1161
## 1-0.7741B 0.7900 0.9898 0.1071
## 1+0.0737B+0.9791B^2 -0.0376+-1.0099i 0.9895 0.2559
## 1-0.8418B+0.9790B^2 0.4299+-0.9147i 0.9894 0.1801
## 1+0.7624B+0.9788B^2 -0.3895+-0.9327i 0.9893 0.3130
## 1+0.3629B+0.9777B^2 -0.1856+-0.9941i 0.9888 0.2794
## 1+0.3268B+0.9775B^2 -0.1672+-0.9975i 0.9887 0.2764
## 1-0.7733B+0.9745B^2 0.3967+-0.9321i 0.9872 0.1860
## 1+1.8763B+0.9744B^2 -0.9628+-0.3151i 0.9871 0.4497
## 1+0.1190B+0.9742B^2 -0.0611+-1.0113i 0.9870 0.2596
## 1+0.9222B -0.9478 0.9864 0.4423
## 1+0.9222B -0.9478 0.9864 0.4423
## 1+1.1153B+0.9727B^2 -0.5733+-0.8363i 0.9863 0.3456
## 1+1.9286B+0.9713B^2 -0.9928+-0.2095i 0.9855 0.4669
## 1-0.6033B 0.6218 0.9850 0.1451
## 1-0.6042B 0.6229 0.9849 0.1449
## 1-0.7066B+0.9698B^2 0.3643+-0.9478i 0.9848 0.1916
## 1+1.2326B+0.9687B^2 -0.6362+-0.7922i 0.9842 0.3577
## 1+1.2884B+0.9685B^2 -0.6651+-0.7682i 0.9841 0.3636
## 1-0.1965B+0.9685B^2 0.1015+-1.0111i 0.9841 0.2341
## 1-0.4304B+0.9658B^2 0.2228+-0.9928i 0.9828 0.2149
## 1-0.6153B 0.6377 0.9823 0.1423
## 1+1.5442B+0.9642B^2 -0.8007+-0.6292i 0.9820 0.3940
## 1+1.8877B+0.9641B^2 -0.9790+-0.2808i 0.9819 0.4555
## 1-0.6150B 0.6390 0.9811 0.1422
## 1+0.9789B -1.0187 0.9803 0.4916
## 1+0.9788B -1.0189 0.9801 0.4916
## 1+0.9718B -1.0220 0.9751 0.4868
## 1+0.9718B -1.0220 0.9751 0.4868
## 1+0.6745B+0.9500B^2 -0.3550+-0.9626i 0.9747 0.3062
## 1-0.4410B+0.9489B^2 0.2324+-0.9999i 0.9741 0.2137
## 1+0.7812B+0.9430B^2 -0.4142+-0.9428i 0.9711 0.3159
## 1+1.7627B+0.9383B^2 -0.9393+-0.4283i 0.9686 0.4319
## 1-0.6968B 0.7471 0.9657 0.1217
## 1-0.6952B 0.7476 0.9643 0.1219
##
## #Transform data with “seasonal difference”, which appears to be stationary, we'll find a stationary model for this realization
x=data$gdp.change
# difference the data
d1=artrans.wge(x,phi.tr=1)
#we try seasonal model with s=27
d1.27=artrans.wge(d1,phi.tr = c(rep(0,28),1))
#Use tswge to model the transformed data, d1.27
aic5.wge(d1.27,p=0:5,q=0:5)## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of aic## p q aic
## 31 5 0 3.249695
## 21 3 2 3.280678
## 2 0 1 3.296534
## 20 3 1 3.307900
## 33 5 2 3.320749# aic picks an ARMA(5,0) as the first choice#In order to see if a lower order model could satisfactorily model the data, we use BIC
aic5.wge(d1.27,p=0:5,q=0:5,type='bic')## ---------WORKING... PLEASE WAIT...
##
##
## Five Smallest Values of bic## p q bic
## 2 0 1 3.323773
## 31 5 0 3.331413
## 21 3 2 3.362396
## 20 3 1 3.375998
## 7 1 0 3.382539#BIC picks ARMA(5,0) as the second choice,We decide to use the ARMA(5,0) model chosen by AIC and BIC# AIC and BIC picks p=1,q=5
fitaruma=arima(data$gdp.change,order=c(5,1,0),xreg=cbind(data$GDP.per.capita,data$Income.receipt,data$gross.income))
fitaruma##
## Call:
## arima(x = data$gdp.change, order = c(5, 1, 0), xreg = cbind(data$GDP.per.capita,
## data$Income.receipt, data$gross.income))
##
## Coefficients:
## ar1 ar2 ar3 ar4 ar5
## -0.3347 -0.0760 -0.1244 -0.1650 -0.1641
## s.e. 0.0801 0.0793 0.0707 0.0671 0.0638
## cbind(data$GDP.per.capita, data$Income.receipt, data$gross.income)1
## -0.0082
## s.e. 0.0015
## cbind(data$GDP.per.capita, data$Income.receipt, data$gross.income)2
## 0.0045
## s.e. 0.0107
## cbind(data$GDP.per.capita, data$Income.receipt, data$gross.income)3
## 0.0177
## s.e. 0.0036
##
## sigma^2 estimated as 12.25: log likelihood = -777.6, aic = 1573.2AIC(fitarima) ## [1] 1572.964#We estimate the parameters of the ARMA(5,0) model using est.ar.wge
x_27_arma=est.ar.wge(d1.27, p=5)##
## Coefficients of Original polynomial:
## -0.4983 -0.2426 -0.2040 -0.2243 -0.3041
##
## Factor Roots Abs Recip System Freq
## 1+0.8053B -1.2418 0.8053 0.5000
## 1+0.7199B+0.6155B^2 -0.5848+-1.1326i 0.7845 0.3259
## 1-1.0268B+0.6135B^2 0.8369+-0.9642i 0.7833 0.1362
##
##
##
## Coefficients of Original polynomial:
## -0.4983 -0.2426 -0.2040 -0.2243 -0.3041
##
## Factor Roots Abs Recip System Freq
## 1+0.8053B -1.2418 0.8053 0.5000
## 1+0.7199B+0.6155B^2 -0.5848+-1.1326i 0.7845 0.3259
## 1-1.0268B+0.6135B^2 0.8369+-0.9642i 0.7833 0.1362
##
## ljung.wge(x_27_arma$res)## Obs -0.02907963 -0.04995584 -0.05843614 -0.07108814 -0.05051819 -0.08992947 -0.07447575 -0.1034145 0.03947868 0.1472326 0.0773277 -0.07160113 -0.2103733 0.06884934 -0.1227055 0.04381979 0.06580372 -0.04145771 -0.02393789 0.1137795 0.06583719 -0.001061071 -0.09334691 0.0520115## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 24
##
## $chi.square
## [1] 48.38316
##
## $df
## [1] 24
##
## $pval
## [1] 0.002262275ljung.wge(x_27_arma$res, K = 48)## Obs -0.02907963 -0.04995584 -0.05843614 -0.07108814 -0.05051819 -0.08992947 -0.07447575 -0.1034145 0.03947868 0.1472326 0.0773277 -0.07160113 -0.2103733 0.06884934 -0.1227055 0.04381979 0.06580372 -0.04145771 -0.02393789 0.1137795 0.06583719 -0.001061071 -0.09334691 0.0520115 0.03694837 0.01176635 0.01150009 0.1051457 -0.34304 -0.08880146 0.08546404 0.03038385 -0.0635886 0.03251928 0.02566841 0.02564567 0.05714387 -0.001871735 -0.1185075 -0.06928661 -0.02930197 0.2215155 -0.05000339 -0.01698825 0.08460033 0.001409524 0.07677338 0.04896264## $test
## [1] "Ljung-Box test"
##
## $K
## [1] 48
##
## $chi.square
## [1] 122.2092
##
## $df
## [1] 48
##
## $pval
## [1] 2.131004e-08f_x27arma = fore.aruma.wge(data$gdp.change,s = 27,phi =x_27_arma$phi,n.ahead =12,limits = F, lastn = T)
f_x27armashort = fore.aruma.wge(data$gdp.change,s = 27,phi =x_27_arma$phi,n.ahead =2,limits = F, lastn = T)
ASE_x27arma= mean((data$gdp.change[(length(x)-11):length(x)] - f_x27arma$f)^2)
ASE_x27arma## [1] 6.365055ASE_x27arma_short= mean((data$gdp.change[(length(x)-1):length(x)] - f_x27armashort$f)^2)
ASE_x27arma_short## [1] 14.18755#Compare Spectral Densities
sims = 5
SpecDen = parzen.wge(data$gdp.change, plot = "FALSE")
plot(SpecDen$freq,SpecDen$pzgram, type = "l", lwd = 6)
for( i in 1: sims)
{
SpecDen4 = parzen.wge(gen.aruma.wge(297, phi = x_27_arma$phi,s=27, plot ="FALSE"), plot = "FALSE")
lines(SpecDen4$freq,SpecDen4$pzgram, lwd = 2, col = "red")
}
#Compare ACFs
sims = 5
ACF = acf(data$gdp.change, plot = "FALSE")
plot(ACF$lag ,ACF$acf , type = "l", lwd = 6)
for( i in 1: sims)
{
ACF5 = acf(gen.aruma.wge(297, phi = x_27_arma$phi,s=27, plot = "FALSE"), plot = "FALSE")
lines(ACF5$lag ,ACF5$acf, lwd = 2, col = "red")
}
Multivariate analysis: VAR Model
data2 = data[1:292,]
gdpVar1 =VAR(cbind(data2$gdp.change,data2$GDP.per.capita,data2$Income.receipt), type="both", lag.max = 10)## Warning in VAR(cbind(data2$gdp.change, data2$GDP.per.capita, data2$Income.receipt), : No column names supplied in y, using: y1, y2, y3 , instead.AIC(gdpVar1) #7177.838## [1] 7342.273preds_short = predict(gdpVar1,n.ahead = 2)
ASE_var1_st = mean((data$gdp.change[291:292] - preds_short$fcst$y1[,1])^2)
ASE_var1_st #ASE for VAR model is 9.952455## [1] 18.07533plot(seq(1,292,1), data$gdp.change[1:292], type = "l",xlim = c(0,292), ylab = "gdp change", main = "Short_term gdp change Forecast")
lines(seq(291,292,1), preds_short$fcst$y1[,1], type = "l", col = "red")
# VAR Model long-term forecast last 24 with much less ASE and AIC than short-term
data2 = data[1:268,]
gdpVar1 = VAR(cbind(data2$gdp.change,data2$GDP.per.capita,data2$Income.receipt), type="both", lag.max = 10)## Warning in VAR(cbind(data2$gdp.change, data2$GDP.per.capita, data2$Income.receipt), : No column names supplied in y, using: y1, y2, y3 , instead.AIC(gdpVar1) ## [1] 6623.122preds = predict(gdpVar1,n.ahead = 24)
ASEvar_lt=mean((data$gdp.change[269:292] - preds$fcst$y1[,1])^2)
ASEvar_lt ## [1] 3.989787plot(seq(1,292,1), data$gdp.change[1:292], type = "l",xlim = c(0,292), ylab = "gdp change", main = "Long_term gdp change Forecast")
lines(seq(269,292,1), preds$fcst$y1[,1], type = "l", col = "red")
Multivariate analysis:Neural network
# MLP fit with 5 hidden nodes and 20 repetitions.
gdp.change=ts(data2$gdp.change)
GDP.per.capita=ts(data2$GDP.per.capita)
Income.receipt=ts(data2$Income.receipt)
xVar=data.frame(GDP.per.capita,Income.receipt)
set.seed(2)
fit.mlp=mlp(gdp.change,xreg=xVar)
fit.mlp## MLP fit with 5 hidden nodes and 20 repetitions.
## Univariate lags: (1,2,3,4)
## 2 regressors included.
## - Regressor 1 lags: (2,4)
## - Regressor 2 lags: (2,3)
## Forecast combined using the median operator.
## MSE: 7.5201.## MLP fit with 5 hidden nodes and 20 repetitions, forecast combined using the median operator, MSE=7.08
xVar1=data.frame(GDP.per.capita=ts(data$GDP.per.capita),Income.receipt=ts(data$Income.receipt))
plot(fit.mlp)
## short-term forecast
fore.mlp_short = forecast(fit.mlp, h=2, xreg = xVar1)
plot(fore.mlp_short)
ASE_nn_st = mean((data$gdp.change[291:292] - fore.mlp_short$mean)^2)
ASE_nn_st ## [1] 15.1618#Plot short_term forecast
plot(seq(1,292,1), data$gdp.change, type = "l",xlim = c(0,295), ylab = "gdp change", main = "Short_term GDP.Change Forecast from MLP")
lines(seq(291,292,1), fore.mlp_short$mean, type = "l", col = "red")
## long-term forecast
fore.mlp = forecast(fit.mlp, h=24, xreg = xVar1)
plot(fore.mlp)
ASE_nn_lt= mean((data$gdp.change[269:292] - fore.mlp$mean)^2)
ASE_nn_lt ## [1] 6.928138#Plot long_term forecast
plot(seq(1,292,1), data$gdp.change, type = "l",xlim = c(0,295), ylab = "gdp change", main = "Long_term GDP.Change Forecast from MLP")
lines(seq(269,292,1), fore.mlp$mean, type = "l", col = "red")
Ensemble model
# long term ensemble
ensemble = (preds$fcst$y1[,1] + fore.mlp$mean)/2
#Plot
plot(seq(1,292,1), data$gdp.change, type = "l",xlim = c(0,292), ylab = "gdp change", main = "Ensemble model long-term Forecast")
lines(seq(269,292,1), y=ensemble, type = "l", col = "RED")
ASE_ensemble = mean((data$gdp.change[269:292] - ensemble)^2)
ASE_ensemble ## [1] 4.783706## short term ensemble model
ensemble = (preds_short$fcst$y1[,1] + fore.mlp_short$mean)/2
#Plot
plot(seq(1,292,1), data$gdp.change, type = "l",xlim = c(0,292), ylab = "gdp change", main = "Ensemble model GDP.Change Forecast")
lines(seq(291,292,1), y=ensemble, type = "l", col = "RED")
ASE_ensemble = mean((data$gdp.change[291:292] - ensemble)^2)
ASE_ensemble #4.8## [1] 16.25906Model Comparison
#For the short-term forecast, we may see ARIMA(5,1,0) with s=27 shows better performance with lowest ASE 14.18755;
#For the long-term forecast, the ensemble model shows the better performance with the lowest ASE 4.78.
print(' *ARMA(4,3) Short-term ASE 37.968, Long-term ASE 6.879' )## [1] " *ARMA(4,3) Short-term ASE 37.968, Long-term ASE 6.879"print(' *ARIMA(5,1) Short-term ASE 35.09, Long-term ASE 6.515' )## [1] " *ARIMA(5,1) Short-term ASE 35.09, Long-term ASE 6.515"print(' *ARUMA(5,1,0) Short-term ASE 14.187, Long-term ASE 6.365' )## [1] " *ARUMA(5,1,0) Short-term ASE 14.187, Long-term ASE 6.365"print(' *VAR Short-term ASE 25.895, Long-term ASE 5.15' )## [1] " *VAR Short-term ASE 25.895, Long-term ASE 5.15"print(' *MLP Short-term ASE 15.16, Long-term ASE 6.27' )## [1] " *MLP Short-term ASE 15.16, Long-term ASE 6.27"print(' *Ensemble Short-term ASE 16.259, Long-term ASE 4.78' )## [1] " *Ensemble Short-term ASE 16.259, Long-term ASE 4.78"Summary
- GDP data is very difficult to predict, and models are not able to capture variance in this data.
- For the short-term forecast, we may see ARIMA(5,1,0) with s=27 shows better performance with lowest ASE 14.18755;
- For the long-term forecast, the ensemble model shows the better performance with the lowest ASE 4.78.
- After all, we can’t simply say which model is better than the other as they perform differently in the different periods.